Matrix

martix parentheses

\begin{pmatrix}
a & b \\
c & d
\end{pmatrix}

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If you want to enclose the matrix in parentheses, you can use the "pmatrix" (p from parentheses +matrix). If you use just "matrix" without adding "p," the matrix will be displayed without parentheses.

martix brackets

\begin{bmatrix}
a & b \\
c & d
\end{bmatrix}

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To enclose a matrix in brackets, you can use the "bmatrix" (b from brackets +matrix). However, if you want the brackets to be curly brackets, you can use the "Bmatrix", with a capital "B".

martix vertical lines

\begin{vmatrix}
a & b \\
c & d
\end{vmatrix}

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To enclose a matrix in vertical lines, you can use the "vmatrix" (v from vertical lines +matrix). However, if you want the vertical lines to be double lines, you can use the "Vmatrix", with a capital "V".

transposed matrix

A^{ \mathrm{ T } }

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To represent the transpose of a matrix, you can use a Roman font uppercase "T" in the top right corner of the matrix.

transposed matrix 2

{}^t \! A

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It is also common to represent the transpose of a matrix by writing a lowercase "t" in the top left corner of the matrix.

dimension

\dim

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The notation "dimension" is derived from the dimension.

matrix rank

\mathrm{ rank } A

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trace

\mathrm{ Tr } A

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determinant

\mathrm{ det }A

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determinant sample

\begin{eqnarray}
\mathrm{ det }A
 = | A |
 = \begin{vmatrix} a & b \\ c & d \end{vmatrix}
 = ad - bc
\end{eqnarray}

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martix large

\begin{pmatrix}
  a & b & c \\
  d & e & f \\
  g & h & I
\end{pmatrix}

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Using the "pmatrix" environment, you can write a 3x3 matrix.

martix large

\begin{eqnarray}
\left(
  \begin{array}{ccc}
    a & b & c \\
    d & e & f \\
    g & h & i
  \end{array}
\right)
\end{eqnarray}

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Using the "array" environment, you can write a matrix. The "ccc" is for center alignment.

right alignment matrix

\begin{eqnarray}
\left(
  \begin{array}{rrr}
    111 & 111 & 111 \\
    22 & 0.2 & -2 \\
    3 & 3 & 3
  \end{array}
\right)
\end{eqnarray}

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Using the "array" environment, you can write a matrix with right alignment for each value by specifying "rrr".

mxn matrix

\begin{eqnarray}
A = \left(
  \begin{array}{cccc}
    a_{ 11 } & a_{ 12 } & \ldots & a_{ 1n } \\
    a_{ 21 } & a_{ 22 } & \ldots & a_{ 2n } \\
    \vdots & \vdots & \ddots & \vdots \\
    a_{ m1 } & a_{ m2 } & \ldots & a_{ mn }
  \end{array}
\right)
\end{eqnarray}

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It is an example of representing an m x n matrix using multiple dots.

block matrix

\begin{eqnarray}
\left(
  \begin{array}{cc|cc}
    a & b & 0 & 0 \\
    c & d & 0 & 0 \\
    \hline
    x & y & 1 & 0 \\
    z & w & 0 & 1 \\
  \end{array}
\right)
\end{eqnarray}

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Using "|" like the "cc|cc", you can draw vertical lines between columns. The "\hline" command can be used to draw horizontal lines.

Jordan block

\begin{eqnarray}
\begin{pmatrix}
  \lambda & 1 &   &  & 0 \\
    & \lambda & 1 &   &   \\
    &   & \ddots & \ddots &   \\
    &   &   & \lambda & 1  \\
  0 &   &   &   & \lambda
\end{pmatrix}
\end{eqnarray}

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cofactor

\begin{eqnarray}
& & (-1)^{ i+j } \times \\[5pt]
& & \quad
\begin{vmatrix}
  a_{1,1} & \ldots & a_{1,j-1} & a_{1,j+1} & \ldots & a_{1,n} \\
  \vdots & \ddots & \vdots & \vdots & \ddots & \vdots \\
  a_{i-1,1} & \ldots & a_{i-1, j-1} & a_{i-1, j+1} & \ldots & a_{i-1, n} \\
  a_{i+1,1} & \ldots & a_{i+1, j-1} & a_{i+1, j+1} & \ldots & a_{i+1, n} \\
  \vdots & \ddots & \vdots & \vdots & \ddots & \vdots \\
  a_{n,1} & \ldots & a_{n, j-1} & a_{n, j+1} & \ldots & a_{n, n}
\end{vmatrix}
\end{eqnarray}

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